Petrie conjecture

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This page has not been refereed. The information given here might be incomplete or provisional.

1 Problem

If a compact Lie group ${{Stub}} == Problem == <wikitex>; If a compact Lie group $G$ acts smoothly and non-trivially on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular? In the case where $M$ is [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]], Petrie {{cite|Petrie1972}} restricted his attention to smooth actions of the Lie group $S^1 = \{z \in \mathbb{C} | |z| = 1\}$, and posed the following conjecture. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth n$-manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., $$p(M) = (1+x^2)^{n+1}$$ for a generator $x \in H^2(M; \mathbb{Z})$. {{endthm}} </wikitex> == Progress to date == <wikitex>; As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However it has been proven in the following special cases. * Petrie {{cite|Petrie1973}} has verified his conjecture under the assumption that the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$. * If $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. * According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$. * Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and {{cite|Masuda1981}}. * By the work of {{cite|Dessai2002}}, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$. * Similarly, the work of {{cite|Dessai&Wilking2004}} confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$ and the action of $S^1$ on $M$ extends to a smooth action of $T^k$ for $k \geq 1$. </wikitex> == Further discussion == <wikitex>; A symplectic version of the Petrie conjecture is discussed in the article of {{cite|Tolman2010}}. </wikitex> == References == {{#RefList:}}  [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts smoothly and non-trivially on a closed smooth manifold $M$ , what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular?

In the case where $M$ is homotopy equivalent to $\CP^n$ , Petrie [Petrie1972] restricted his attention to smooth actions of the Lie group $S^1 = \{z \in \mathbb{C} | |z| = 1\}$ , and posed the following conjecture.

Conjecture 1.1 [Petrie1972].

Suppose that $M$ $M$ is a closed smooth $2n$ $2n$ -manifold homotopy equivalent to $\CP^n$ $\CP^n$ and that $S^1$ $S^1$ acts smoothly and non-trivially on $M$ $M$ . Then the total Pontrjagin class $p(M)$ $p(M)$ of $M$ $M$ agrees with that of $\CP^n$ $\CP^n$ , i.e.,

$\displaystyle p(M) = (1+x^2)^{n+1}$ $\displaystyle p(M) = (1+x^2)^{n+1}$

for a generator $x \in H^2(M; \mathbb{Z})$ .

2 Progress to date

As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However it has been proven in the following special cases.

Petrie [Petrie1973] has verified his conjecture under the assumption that the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$ .
If $\dim M \leq 8$ , the statement is true by the work of [Dejter1976] and [James1985].
According to [Hattori1978], the conjecture is also true if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$ .
Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981].
By the work of [Dessai2002], the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$ .
Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$ and the action of $S^1$ on $M$ extends to a smooth action of $T^k$ for $k \geq 1$ .